Polynomial-time Tractable Problems over the $p$-adic Numbers
Arno Fehm, Manuel Bodirsky
TL;DR
The work resolves the complexity of satisfiability for linear systems augmented with $v_p$-valuation constraints over $\mathbb{Q}_p$ and $\mathbb{Z}_p$, producing a complete picture for fixed primes: ${\mathbb Z}_3$ and ${\mathbb Q}_3$ yield NP-completeness, while ${\mathbb Z}_2$ and ${\mathbb Q}_2$ lie in ${\rm P}$. It also delivers polynomial-time algorithms for constraints of the form $v_p(x) \le c$ and $v_p(x) \ge c$, and extends these methods to decide satisfiability when combining valuation constraints with linear inequalities over $\mathbb{Q}$ across multiple primes via the approximation theorem for absolute values. The approach leverages constraint satisfaction frameworks, primitive positive interpretations, and Q_p-to-Q reductions, enabling transfer of hardness and tractability results between $\mathbb{Q}_p$ and $\mathbb{Q}$ and across primes. An open problem remains about the existence of a polynomial-time algorithm for weak linear inequalities with coefficients $2^c$ (binary $c$), a question tied to mean-payoff games, highlighting both practical tractability gains and intriguing computational boundaries.
Abstract
We study the computational complexity of fundamental problems over the $p$-adic numbers ${\mathbb Q}_p$ and the $p$-adic integers ${\mathbb Z}_p$. Guépin, Haase, and Worrell proved that checking satisfiability of systems of linear equations combined with valuation constraints of the form $v_p(x) = c$ for $p \geq 5$ is NP-complete (both over ${\mathbb Z}_p$ and over ${\mathbb Q}_p$), and left the cases $p=2$ and $p=3$ open. We solve their problem by showing that the problem is NP-complete for ${\mathbb Z}_3$ and for ${\mathbb Q}_3$, but that it is in P for ${\mathbb Z}_2$ and for ${\mathbb Q}_2$. We also present different polynomial-time algorithms for solvability of systems of linear equations in ${\mathbb Q}_p$ with either constraints of the form $v_p(x) \leq c$ or of the form $v_p(x)\geq c$ for $c \in {\mathbb Z}$. Finally, we show how our algorithms can be used to decide in polynomial time the satisfiability of systems of (strict and non-strict) linear inequalities over ${\mathbb Q}$ together with valuation constraints $v_p(x) \geq c$ for several different prime numbers $p$ simultaneously.